Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
C.9.1 - Differential Equations Calculus - Santowski 5/24/2017 Calculus - Santowski 1 Lesson Objectives • 1. Become familiar with a definition of and terminology involved with differential equations • 2. Solve differential equations with and without initial conditions • 2. Apply differential equations in a variety of real world applications 5/24/2017 Calculus - Santowski 2 (A) Definitions • A differential equation is any equation which contains derivatives dy x sin y • Ex: dx • Ex: Newton’s second law of motion can also be rewritten as a differential equation: • F = ma which we can rewrite as dv d 2s Fm or F m 2 dt dt 5/24/2017 Calculus - Santowski 3 (A) Definitions • The order of a differential equation refers to the largest derivative present in the DE • A solution to a differential equation on a given interval is any function that satisfies the differential equation in the given interval 5/24/2017 Calculus - Santowski 4 (B) Example 3 2 • Show that y(x) x is a solution to the 2 4 x y 12xy 3y 0 second order DE for x > 0 • Why is the restriction x > 0 necessary? • Are the equations listed below also solutions?? 1 3 3 y(x) x 2 or y(x) 9x 5/24/2017 2 or y(x) 9x Calculus - Santowski 2 7x 1 2 5 (C) Initial Conditions • As we saw in the last example, a DE will have multiple solutions (in terms of many functions that satisfy the original DE) • So to be able to identify or to solve for a specific solution, we can given a condition (or a set of conditions) that will allow us to identify the one single function that satisfies the DE 5/24/2017 Calculus - Santowski 6 (D) Example - IVP 3 2 • Show that y(x) x is a solution to the 2 4 x y 12xy 3y 0 second order DE given the initial conditions of 1 3 y(4) and y (4) 8 64 3 2 • What about the equation y(x) 9x 7x 5/24/2017 Calculus - Santowski 1 2 7 (D) Examples 3 c • Ex 1. Show that y(t) 4 t 2 is the general solution to the DE 2ty 4y 3 is the actual solution to the IVP • Ex 2. What and y(1) 4 2ty 4y 3 5/24/2017 Calculus - Santowski 8 (E) Further Examples • Ex 1. What is the solution to the first order differential equation (FODE) dy 2x dx • Sketch your solution on a graph. • Then solve the IVP given the initial condition of (-1,1) 5/24/2017 Calculus - Santowski 9 (E) Further Examples • Ex 2. Solve the IVP dy 1 sin x and e,1 cos(e) dx x • How would you verify your solution? 5/24/2017 Calculus - Santowski 10 (F) Motion Examples • If a body moves from rest so that its velocity is proportional to the time elapsed, show that v2 = 2as where s = displacement, t is time and v is velocity and a is a constant (representing ….?) 5/24/2017 Calculus - Santowski 11 (F) Motion Examples • A particle is accelerated in a line so that its velocity in m/s is equal to the square root of the time elapsed, measured in seconds. How far does the particle travel in the first hour? 5/24/2017 Calculus - Santowski 12 (F) Motion Examples • A stone is tossed upward with a velocity of 8 m/s from the edge of a cliff 63 m high. How long will it take for the stone to hit the ground at the foot of the cliff? 5/24/2017 Calculus - Santowski 13 (F) In Class Worksheets • We will work through selected questions from handouts from several other Calculus textbooks 5/24/2017 Calculus - Santowski 14 (G) DE on TI-89 QuickTime™ and a TIFF (Uncomp resse d) de com press or are nee ded to s ee this picture. QuickTime™ and a TIFF (Uncomp resse d) de com press or are nee ded to s ee this picture. QuickTime™ and a TIFF (Uncomp resse d) de com press or are nee ded to s ee this picture. QuickTime™ and a TIFF (Uncomp resse d) de com press or are nee ded to s ee this picture. 5/24/2017 Calculus - Santowski 15 (G) DE on TI-89 • So the last slides have shown how to solve DE on the TI-89 as well as preparing graphs (which we will explore in more detail in a future lesson) 5/24/2017 QuickTime™ and a TIFF (Uncomp resse d) de com press or are nee ded to s ee this picture. Calculus - Santowski 16